Wilson Lines

When compactifying a coordinate on a circle of radius R, $x^d\sim x^d+2\pi R$, we will introduce a classical background gauge field

$A_d=-\frac{1}{2\pi R}\text{diag}({\theta }_1,\dots {\theta }_N)$.

This background is flat; it has a vanishing field strength and hence can be set locally to zero by a gauge transformation. We can only shift ${\theta }_j\sim {\theta }_j+2\pi n_j$, for $n\in \mathbb{Z}$ by a globally defined gauge transformation $A_d\to f^{-1}(-i{\partial }_d+A_d)f$ with $f=e^{-in{x}^d/R}$. Then,

$A_d\mapsto -\frac{n}{R}+A_d$

$-\frac{1}{2\pi R}\theta \mapsto -\frac{1}{2\pi R}\theta -\frac{n}{R}$

$\theta \mapsto \theta +2\pi n$.

Thus ${\theta }_j$ are naturally periodic. This is known as the Aharonov-Bohm effect.

The background that we chose spontaneously breaks $U(N)\to U(1{)}^N$, which is its maximal torus.

The non-abelian version of the Aharonov-Bohm effect means a particle transforming in the fundamental representation will undergo a non-trivial transformation upon transportation around the circle,

$P\text{exp}i\oint d{x}^d{A}_d=\text{diag}(e^{-i{\theta }_1},\dots ,e^{-i{\theta }_N})$.

This is known as holonomy of the gauge field. A point particle in state $|j⟩$ will pick up a phase $e^{-i{\theta }_j}$ around the circle.

This modifies the quantisation of the KK momentum. This arose from the condition $e^{2\pi R{p}^d}=1$, representing the fact that translation by $x^d\to x^d+2\pi R$ is trivial. The Wilson line modifies this condition. The $ij$-strings must satisfy

$e^{2\pi iR{p}^d}{e}^{-i({\theta }_i-{\theta }_j)}=1$

and therefore

$p^d=\frac{{\theta }_i-{\theta }_j+2\pi n}{2\pi R}$.

For $ii$-strings, it reduces to our previous result of $p^d=n/R$.